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Math Help - Projection of line on plane

  1. #1
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    Projection of line on plane

    Given \pi: \begin{cases}l_1: \frac{x-2}{4}\ = \frac{y-1}{2}\ = \frac{z+5}{-4} &\\l_2: \frac{x+4}{-2}\ = \frac{y+1}{0}\ = \frac{z}{1} & \end{cases} and the point M=(1,2,3) outside the plane. Find the projection of the line M_1P on plane \pi where P is the intersection point of lines l_1, l_2.

    All I can do is that I can find the equation of plane \pi but don't have any idea what to do next.

    I need your help.

    Thank you.
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  2. #2
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    Quote Originally Posted by patzer View Post
    Given \pi: \begin{cases}l_1: \frac{x-2}{4}\ = \frac{y-1}{2}\ = \frac{z+5}{-4} &\\l_2: \frac{x+4}{-2}\ = \frac{y+1}{0}\ = \frac{z}{1} & \end{cases} and the point M=(1,2,3) outside the plane. Find the projection of the line M_1P on plane \pi where P is the intersection point of lines l_1, l_2.
    First a comment: No competent textbook or instructor would write a like this one did. It should be l_2: \frac{x+4}{-2}\ = \frac{z}{1};~y=-1. .

    Now I will not do the work for you.
    First show the line intersect in P.
    The directions of the lines are D_1:<4,2,-4>~\&~D_2:<-2,0,1>.
    The plane containing P and with normal N=D_1\times D_2 is \pi.
    The projection is the intersection of the plane containing P with normal  \overrightarrow {PM}  \times N with plane \pi.
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